nLab Sandbox

$$\mathrm{d} e^i \;=\; \epsilon^{i j k } e_j e_k$$

$$\omega^{i j} \;\equiv\; - \epsilon^{i j k} e_k$$

$$\begin{array}{l} \mathrm{d} e^i - \omega^{i j} e_j \\ \;=\; \epsilon^{i j k } e_j e_k + \epsilon^{i j k} e_k\, e_j \\ \;=\; 0 \end{array}$$

$$\begin{array}{l} \mathrm{d} \omega^{i j} \\ \;=\; - \epsilon^{i j k} \epsilon_{k l m} e^l \, e^m \\ \;=\; -2 \delta^{ i j }_{l m} e^{l}\, e^m \\ \;=\; -2 e^{i}\, e^{j} \end{array}$$

$$\begin{array}{l} \omega^{i k} \, \omega_{k}{}^j \\ \;=\; \epsilon^{i k l} \epsilon_{k j m} e_l \, e^m \\ \;=\; -2\delta^{i l}_{j m} e_l \, e^m \\ \;=\; - e^i \, e^j \end{array}$$

$$\begin{array}{l} R^{i j} \\ \;=\; \mathrm{d}e^i - \omega^{i k}\, \omega_k{}^j \\ \;=\; -2 e^{i}\, e^{j} + e^i \, e^j \\ \;=\; - e^{i}\, e^{j} \end{array}$$

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